Let us call this as the predicted observation. The equation for the corrected
state in a Kalman filter is given as
corrected State = predicted state + k(actual observation –- predicted observation)
The difference between the actual observation and the predicted obser-
vation is called the observation innovation. Note that a fraction of the ob-
servation innovation is added as a correction to the predicted state. The
value of this fraction kis known as the Kalman gain. The Kalman-filtering
approach provides a prescription on what would be the most appropriate
value to use for k. This value is decided such that the corrected state has the
least amount of error variance associated with it.
Besides providing the prescription to reconcile the prediction and ob-
servation, Kalman also provided definitive proof that the process is indeed
optimal in the case where the mathematical models of state and observation
are both linear and the errors are drawn from independent Gaussian distri-
butions. We will, of course, not delve into the proofs, but rather try to ex-
plain the basic idea by way of illustrations. With that said, we introduce
some notation and formally list the steps involved in the Kalman-filtering
process.
LetXtdenote the state at time t. Note that the value for the state can
also be a vector; that is, the state has a multidimensional representation. The
mathematical model used to predict the state at time tin a Kalman filter set-
ting is typically of the form
Xt=AXt–1+ut (4.1)whereAis a matrix, XtandXt–1are the state vectors at time tandt– 1, re-
spectively, and utis the error vector that accounts for the impreciseness of
the model. Next we make an observation at time t. Let us call this observa-
tionYt. The measurements made are a linear combination of the state ele-
ments and therefore can be written as
Yt=HXt+vt (4.2)In this scenario the values of the matrices AandHare known. Initially
we make a prediction of the state at time t, knowing all the state informa-
tion up to time t– 1. Let us denote this estimate. The error is measured
as the variance in the case of a single dimensional state and as a covariance
matrix in the case of a multidimensional state.
Let us denote it generally as. Just as in the case of the predicted
value, the measurement also has an error variance/covariance matrix associated
ˆ
Ptt|− 1ˆ
Xtt|− 156 BACKGROUND MATERIAL